We say that a category is algebraically complete when every endofunctor has an initial algebra. Similarly, a category is algebraically cocomplete when every endofunctor has a final coalgebra.
Assuming the Generalized Continuum Hypothesis, the category of sets of cardinality less or equal a certain regular uncountable ordinal, $\mathsf{Set}_\lambda$, is algebraically complete and cocomplete. This is proven, for instance, in Adamek's "On Terminal Coalgebras Derived from Initial Algebras".
What about categories of presheaves valued in $\mathsf{Set}_\lambda$? Is $[\mathbb{C},\mathsf{Set}_\lambda]$, for $\mathbb{C}$ with cardinality of objects and morphisms less or equal than $\lambda$, algebraically complete and cocomplete? Is there any reference for the topic of when do (terminal)initial (co)algebras for endofunctors over presheaf categories exist?
When $\mathbb{C}$ is a discrete category, the same paper explains that $[\mathbb{C},\mathsf{Set}_\lambda]$ is algebraically complete and cocomplete whenever $\lambda > |\mathbb{C}|$. This uses the fact that every object in the category can be written as the coproduct of the functors choosing 1 at some object and 0 everywhere else. However, it seems to me that the same argument cannot be applied when $\mathbb{C}$ is not discrete: for instance, there are many presheaves over the diagram with two parallel arrows that cannot be decomposed as the coproduct of arrows.
I have also found a result saying that accessible endofunctors on locally presentable categories that preserve monics do have a terminal coalgebra (see, for instance, Worrell's "Terminal sequences for accessible endofunctors"). Can we have a stronger result if we restrict to presheaf categories?